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The Art of Conjecturing (Ars Conjectandi). on the Historical Origin of Normal Distribution [Rodowód Rozkładu Normalnego]

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The Art of Conjecturing (Ars Conjectandi). on the Historical Origin of Normal Distribution [Rodowód Rozkładu Normalnego] DIDACTICS OF MATHEMATICS 7(11) The Publishing House of the Wrocław University of Economics Wrocław 2010 Editors Janusz Łyko Antoni Smoluk Referee Marian Matłoka (Uniwersytet Ekonomiczny w Poznaniu) Proof reading Agnieszka Flasińska Setting Elżbieta Szlachcic Cover design Robert Mazurczyk Front cover painting: W. Tank, Sower (private collection) © Copyright by the Wrocław University of Economics Wrocław 2010 PL ISSN 1733-7941 Print run: 200 copies TABLE OF CONTENTS MAREK BIERNACKI Applications of the integral in economics. A few simple examples for first-year students [Zastosowania całki w ekonomii] .......................................................... 5 PIOTR CHRZAN, EWA DZIWOK Matematyka jako fundament nowoczesnych finansów. Analiza problemu na podstawie doświadczeń związanych z uruchomieniem specjalności Master Program Quantitative Asset and Risk Management (ARIMA) [Mathematics as a foundation of modern finance] .............................................................................................. 15 BEATA FAŁDA, JÓZEF ZAJĄC Algebraiczne aspekty procesów ekonomicznych [Algebraical aspects of economics processes] .......................................................................................... 23 HELENA GASPARS-WIELOCH How to teach quantitative subjects at universities of economics in a comprehensible and pleasant way? [Jak uczyć ilościowych przedmiotów na uczelniach ekonomicznych w zrozumiały i przyjemny sposób?] ......................... 33 DONATA KOPAŃSKA-BRÓDKA Wspomaganie dydaktyki matematyki narzędziami informatyki [Information technology supporting mathematical education] ................................................ 49 PATRYCJA KOWALCZYK, WANDA RONKA-CHMIELOWIEC Metody matematyczne w dydaktyce ubezpieczeń na studiach ekonomicznych [Mathematical methods in the didactics of insurance on economic studies] ...... 59 LUDOMIR LAUDAŃSKI The art of conjecturing (Ars Conjectandi). On the historical origin of normal distribution [Rodowód rozkładu normalnego] .................................................... 67 JANUSZ ŁYKO, ANDRZEJ MISZTAL Wpływ zmiany liczby godzin zajęć na wyniki egzaminu z matematyki na kie- runkach ekonomicznych [The impact of changes in the number of hours of classes on exam results in mathematics at the economic faculties] .................... 81 KRZYSZTOF MALAGA Matematyka na usługach mikroekonomii [Mathematics on microeconomics services] .............................................................................................................. 93 WOJCIECH RYBICKI Kilka powodów, dla których opowiadamy studentom ekonomii o macierzach [Some reasons for which we tell students of economics about matrices] ........... 109 ANDRZEJ WILKOWSKI On changing money and the birthday paradox [O rozmienianiu pieniędzy i paradoksie urodzin] ......................................................................................... 127 HENRYK ZAWADZKI Mathematica® na usługach ekonomii [Mathematica® at economics service] ......... 135 DIDACTICS OF MATHEMATICS No. 7 (11) 2010 THE ART OF CONJECTURING (ARS CONJECTANDI)1 ON THE HISTORICAL ORIGIN OF NORMAL DISTRIBUTION Ludomir Laudański Abstract. The paper offers a range of historic investigations regarding the normal distribu- tion, frequently also referred to as the Gaussian distribution. The first one is the Error Analysis, the second one is the Probability Theory with its old exposition called the Theory of Chance. The latter regarded as the essential, despite the fact that the origin of the Error Theory can be associated with Galileo Galilei and his Dialogo sopra i due massimi sistemi del mondo Tolemaico e Copernico. However, the normal distribution regarded in this way was not found before 1808-9 as a result of the combined efforts of Robert Adrain and his Researches Concerning Isotomous Curves on the one hand and Carl F. Gauss and his Theoria motus corporum coelestium in sectionibus conicis Solem ambienitum on the other. While considering the Theory of Chance – it is necessary to acknowledge The Doctrine of Chances of Abraham de Moivre – 1733 and the proof contained in this work showing the normal distribution derived as the liming case of the binomial distribution with the number of Bernoulli trials tending to infinity. Therefore the simplest conclusion of the paper is: the normal distribution should be rather attributed to Abraham de Moivre than to Carl Friedrich Gauss. Keywords: binomial distribution, Errors Theory, normal distribution. Personal statement. One day I was asked what the origin of Gaussian distribution was and whether I may shortly explain its origin? After one month of efforts dedicated to this question I had to confess that my hasty answer was inaccurate and reflected my underestimation of the problem. But in one respect I was right, i.e. in considering two historical approaches leading towards the right answer. Error Analysis exposed in the paper (Heller, Paderta 1974) in which the authors present the following axiomatic assertions to derive the normal distribution (quoted literally): Ludomir Laudański Department of Quantitative Methods in Economics, Rzeszów University of Technology, al. Powstańców Warszawy 8, 35-959 Rzeszów Katowice, Poland. e-mail: [email protected] 1 Reference to a title of an original paper written by Jacob Bernoulli in 1713 means: ―forming an opinion or supposition about (something) on the basis of incomplete information‖. 68 Ludomir Laudański ―1. the probability of occurrence of small random errors is greater than probability of occurrence of big random errors; 2. random errors of the same absolute value but opposite sign are equal- ly probable‖. Developing this approach they arrived at a functional equation which, as you can easily check, meets the normal function. They also took care of appropriate normalization so that it could meet conditions of the probability density functions. The functional expression they got has only a single constant – the variance, which means that the mean is assumed to be zero. The above described result should be confronted with two important re- marks. The first follows a suggestion given in (Juszkiewicz [Ed.] 1977) pointing out that the above stated axioms were first formulated by Galileo Galilei (1564-1642) in his famous Dialogues (Galileo Galilei 1962) while analyzing the problem of the astronomic observation of Nova year 1572 – stated by Salviatti (porte parole of the Author). Though the reader should be warned about extremely verbose character of the book dialogues (in Polish translation it covers pages 301-341) – therefore a significant effort and inquiring mind are required to fish out these axioms from the slowly flow- ing discourse in the Dialogue. This fact reduces merits of Heller and Paderta (Heller, Paderta 1974). Moreover they were completely unaware of the above described Galileo‘s contribution to the field of Error Analysis (quot- ing instead some third-rate contemporary source book on this matter). But there is also a second important remark – this time pointing out to Carl Friedrich Gauss (1777-1854) – whose book Theoria motus corporum coe- lestium in sectionibus conicis Solem ambienitum published in Hamburg in 1809 is considered to be the first book on mathematical treatment of the experimental errors — therefore it is Gauss with whom they also have to share their final result (Gauss 1857). Thanks to the English translation of the Latin Theoria motus… – now accessible via the Internet – it is possible to track – step by step – the entire way in which Gauss obtained his functional equation finally leading to the desirable function of errors (the complete procedure covers pages 249-273, but the resulting functional equation is to be found on p. 258). To the above one has to add yet another finding which, in its part, narrows Gauss contribution to the matter. There was a little known American mathematician – Robert Adrain (1775-1843), who discov- ered that formula probably earlier (than Gauss) and published a paper re- lated to the evaluation of the quantitative observations of the animal species of the sea (Adrain 1803) – and in this paper he also came to the ―bell shaped On the historical origin on normal distribution 69 curve‖. Now we come to an important question: whether the above de- scribed facts really lead to the discovery of the normal curve for the first time? The answer is NO. Therefore there are not so many important reasons to study in detail particular contributors working in this field. Below we shall present instead many more details regarding the other way which leads to the discovery of the normal distribution for the first time. Concluding this paragraph I propose a conclusion that credit for the discovery of the normal curve should not go to Gauss – not questioning his pioneering results in establishing the Error Analysis – he made so many brilliant mathematical discoveries, but in this one he was not the leader but rather a follower. Theory of Chance, my favorite text on history of mathematics, for a long time has been a single volume book by Carl B. Boyer (Boyer 1985). Unfortunately this time it brought nothing but disappointment. Therefore I switched to the easily accessible in Poland books by Soviet historians (Juszkiewicz 1977) and (Майстров 1967) – despite cautious criticism and lack of confidence – nevertheless the passages devoted to the history of Probability and related disciplines I found in general sufficiently well coin- ciding with the other sources listed in Bibliography of this paper – among them with the history
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